Question

Find the domains of definition of the following function:
$y=\sqrt{\frac{1-5x}{7^{-x}-7}}$

Answer 1

case$1$

$1-5x\ge 05x\le 1x\le \frac{1}{5}$

and

$7^{-x}-7>0$

$7^{-x}>7$

$7^{-x-1}>1$

taking $\log$ we get

$(-x-1){\log }_7\left(7\right)>{\log }_7\left(1\right)$

$(-x-1)>0$

$x+1<0$

so, $x<(-1)$

case$2$

$1-5x\le 05x\ge 1x\ge \frac{1}{5}$

and

$7^{-x}-7<0$

$7^{-x}<7$

$7^{-x-1}<1$

taking $\log$ we get

$(-x-1){\log }_7\left(7\right)<{\log }_7\left(1\right)$

$⟹(-x-1)<0$

$⟹x+1>0$

$⟹x>(-1)$

so $x\ge \frac{1}{5}$

thus combiningly, domain is

$f\left(x\right)=(-\infty ,-1)\cup (\frac{1}{5},\infty )$